Digamma Function of One Half

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Theorem

$\displaystyle \map \psi {\frac 1 2} = -\gamma - 2 \ln 2$

where:

$\psi$ denotes the digamma function
$\gamma$ denotes the Euler-Mascheroni constant.


Proof

\(\displaystyle \map \psi {\frac 1 2}\) \(=\) \(\displaystyle -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {\pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }\) Gauss's Digamma Theorem
\(\displaystyle \) \(=\) \(\displaystyle -\gamma - \ln 4\) Cotangent of Half-Integer Multiple of Pi, noting also that the summation is an empty sum
\(\displaystyle \) \(=\) \(\displaystyle -\gamma - 2 \ln 2\) Logarithm of Power

$\blacksquare$